
the wave model in [1]} where the combination of the mass and the damping term produces an exponential decay for \(L^2(\mathbf {H}_n)\)  data and even for \(L^1(\mathbf {H}_n)\)  data;

the wave model in [2]} where the classical damping term produces an exponential decay for \(L^2(\mathbf {H}_n)\)  data and a polynomial decay for \(L^1(\mathbf {H}_n)\)  data;

